where we considered for simplicity \(m_{\pi ^{+}}\simeq m_{\pi ^0}\simeq 137\)  MeV. Hence, for a given photon energy \(\varepsilon ^{\prime }\)  this requires that \(E^{\prime }_p>70\,{\rm TeV}/(\varepsilon ^{\prime }/{\rm keV})\) .
The different functions \(\Phi _i(\eta ,x_i)\)  where obtained in [1]} using SOPHIA based [2]} simulations and were conveniently parameterized there, with the parameters of the fits being tabulated for different values of \(\eta /\eta _0\) .We used a linear interpolation between the tabulated values, and for \(\eta /\eta _0>100\)  we used for it the values of \(\Phi _i(100\eta _0,x)\)  in order to be able to approximately extend the computation up to the highest energies that we considered. These functions are plotted in Fig. REF  for the different neutrino and antineutrino flavors as well as for the photons (although additional photons may result for instance from cascading of secondary electrons).
<FIGURE>